Spatial Scaling of Effective Modulus and Correlation of Deformation Near the Critical Point of Fracturing

Many observations point to the lithosphere being metastable and close to a critical mechanical point. Exercises in modelling deformation, past or present, across subsurface reservoirs need to take account of this criticality in an efficient way. Using a renormalization technique, the spatial scaling of effective elastic modulus is derived for 2-D and 3-D bodies close to the critical point of through-going fracturing. The resulting exponent, dl, of spatial scaling of effective modulus with size, Ldl, takes the values )2.5 and )4.2 in two- and three-dimensional space, respectively. The exponents are compatible with those for scaling of effective modulus with fracture density near the percolation threshold determined by other workers from numerical experiments; the high absolute values are also approximately consistent with empirical data from a) fluctuations in depth of a seismic surface; b) ‘1/k’ scaling of heterogeneities observed in one-dimensional well-log samples; c) spatial correlation of slip displacements induced by water injection. The effective modulus scaling modifies the spatial correlation of components of displacement or strain for a domain close to the critical point of fracturing. This correlation function has been used to geostatistically interpolate components of the strain tensor across subsurface reservoirs with the prime purpose of predicting fracture densities between drilled wells. Simulations of strain distributions appear realistic and can be conditioned to surface depths and observations at wells of fracture-related information such as densities and orientations, welltest permeabilities, changes in well-test permeabilities, etc.